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@@ -54,12 +54,13 @@ account for uncertainty is vital in performance, relibiability, and safety of hi
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A comprehensive uncertainty framework for scientific computation involves a mathematical model,
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through which various input uncertainties are propagated to estimate the uncertainty of an unknown quantity of interest (QoI).
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Real world complex systems (physical or engineered) of industrial significance typically involves input parameters subject to uncertainties of various nature [@oberkampf:2004; @smith:2004]. These input uncertainties often appear as mixed uncertainties. A common example is probability boxes (p-boxes), which effectively represent a set of distributions and thereby capture both aleatory and epistemic uncertainty within a single structure. More generally, mixed uncertainties may involve a mixture of parameters subject to variability (aleatory uncertainty, typically represented by probability distributions), or lack of knowledge (epistemic uncertainty, often represented by intervals), or combinations of these forms, such as p-boxes.
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Real-world complex systems (physical or engineered) of industrial significance typically involves input parameters subject to uncertainties of various nature [@oberkampf:2004; @smith:2004]. These input uncertainties often appear as a mixture of parameters subject to variability (i.e. aleatory uncertainty, typically represented by probability distributions), or lack of knowledge (i.e. epistemic uncertainty, often represented by intervals), or a combination of both (i.e. mixed uncertainty, often represented by probability boxes (p-boxes)), which effectively represent a set of distributions and thereby capture both aleatory and epistemic uncertainty within a unified structure.
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Probability bounds analysis [@springer1979algebra; @williamson1990probabilistic; @ferson:2001] is one of the expressive frameworks proposed to manage uncertainties in an imprecise setting [@beer:2013].
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Software packages have been developed to facilitate the calculations of uncertain quantities, such as *interval arithmetic*[@marco_2022_6205624] and *probability arithmetic*[@gray:2021; @gray:2022]. Collectively, they can be referred to as uncertainty arithmetic [@chen:2025], which provides a straightforward way to compute outcomes from algebraic expressions involving uncertain variables.
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Software packages have been developed to facilitate the calculations of uncertain quantities, such as *interval arithmetic*[@marco_2022_6205624] and *probabilistic arithmetic*[@gray:2021; @gray:2022]. Collectively, they can be referred to as *uncertainty arithmetic*[@chen:2025], which provides a straightforward way to compute outcomes from algebraic expressions involving uncertain variables.
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While these uncertainty-representing objects have the potential to automatically compile non-deterministic subroutines via uncertain primitives, their use faces several challenges. A major one is that code accessibility[^1] of the simulation models or software (e.g. finite-element or computational-fluid-dynamics models) employed to describe the behavior of the system is often not guaranteed, rendering intrusive[^2] use of the above-mentioned uncertainty-representing frameworks infeasible. Often, these models are treated as black-box models in practice.
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While these uncertainty-representing frameworks have the potential to automatically compile non-deterministic subroutines via uncertain primitives, their use faces several challenges. A major one is that code accessibility[^1] of the simulation models or software (e.g. finite-element or computational-fluid-dynamics models) employed to describe the behavior of the system is often not guaranteed, rendering intrusive[^2] use of the above-mentioned uncertainty-representing frameworks infeasible. Often, these models are treated as black-box models in practice.
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Consequently, there is a critical need for software tools capable of performing mixed-uncertainty calculations on black-box models in real-world applications of computational engineering and physics.
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<!-- This would largely restrict the adoption of these tools -->
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<!-- besides known issues such as [dependency problems](https://pyuncertainnumber.readthedocs.io/en/latest/examples/repeated_variable.html) -->
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`pyuncertainnumber` addresses this need by providing the non-intrusive[^3] capability designed to allow generic black-box models to be rigorously propagated in the face of mixed uncertainties.
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This capability significantly boosts versatility for scientific computations through interfacing with many engineering software.
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This capability significantly boosts versatility for scientific computations through interfacing with many engineering software whose internal workings are not known.
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@@ -138,14 +139,14 @@ where $\mathbf{u} \in \mathbb{R}^{n}$ denotes the collection of $n$ uncertain in
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When both aleatory and epistemic uncertainties are present in $\mathbf{u}$, a *nested (double) Monte Carlo* approach can be used for determininsitc models without confounding the two distinct types of uncertainty.
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As illustrated in \autoref{fig:dmc}, Latin-hypercube samples are first drawn from the epistemic interval, conditioned on which aleatory samples are drawn from the aleatoric probability distributions.
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As illustrated in \autoref{fig:dmc} with the examplar function B (see \autoref{fig:two_functions}), Latin-hypercube samples are first drawn from the epistemic interval, conditioned on which aleatory samples are drawn from the aleatoric probability distributions.
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Propagate these samples, which are visually denoted as rug ticks alongside the abscissa, through the computational model results in an ensemble of CDF (cumulative distribution function) of the QoI whereby a final p-box is obtained as the envelope.
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Each CDF (orange color) correponds to an epistemic sample.
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To scale to a more realistic setting, \autoref{fig:imc} illustrates the workflow of *interval Monte Carlo* method where a mixture of aleatory, epistemic, and mixed uncertainty parameters are present, and a certain copula is specified denoting the dependency structure. Correlated samples in the uniform space from the copula, visually denoted as rug ticks alongside the probability axis, are converted to physical space through alpha-cuts. Interval propagation (see [the last section](#interval-propagation-in-a-non-intrusive-manner)) then does the heavy lifting in which scalar values can be considered as degenerate intervals. As a result, the response QoI in \autoref{fig:imc} is then obtained as a p-box shown in gray. In contrast, a pinched response, obtained from propagating pinched input variables (e.g. a p-box is pinched into a distribution and an interval is pinched into a scalar), is also shown as a comparison. Importantly, the pinched result being enclosed in the p-box manifests a critical feature of probability bounds analysis which yields results that are guaranteed to encolse all possible distributions of the ouput so long as the input p-boxes were all sure to enclose their respective distributions.
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To scale to a more realistic setting, \autoref{fig:imc} illustrates the workflow of *interval Monte Carlo* method using the examplar function A where a mixture of aleatory, epistemic, and mixed uncertainty parameters are present, plus a certain copula is specified denoting the dependency structure. Correlated samples in the uniform space from the copula, visually denoted as rug ticks alongside the probability axis, are converted to physical space through alpha-cuts. Interval propagation (see [the last section](#interval-propagation-in-a-non-intrusive-manner)) then does the heavy lifting in which scalar values can be considered as degenerate intervals. As a result, the response QoI in \autoref{fig:imc} is then obtained as a p-box shown in gray. In contrast, a pinched response, obtained from propagating pinched input variables (e.g. a p-box is pinched into a distribution and an interval is pinched into a scalar), is also shown as a comparison. Importantly, the pinched result being enclosed in the p-box manifests a critical feature of probability bounds analysis which yields results that are guaranteed to encolse all possible distributions of the ouput so long as the input p-boxes were all sure to enclose their respective distributions.
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@@ -158,7 +159,7 @@ To scale to a more realistic setting, \autoref{fig:imc} illustrates the workflow
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It is evident that many computational tasks in engineering and physics rely on complex numerical methods in which the model functions are black boxes.
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This makes uncertainty arithmetic difficult to manage because the model code is inaccessible.
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By providing methods supporting generic blackbox models, `pyuncertainnumber` enables rigorous uncertainty analysis for real-world scenarios of mixed uncertainties and partial knowledge.
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By providing methods supporting generic black-box models, `pyuncertainnumber` enables rigorous uncertainty analysis for real-world scenarios of mixed uncertainties and partial knowledge.
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This non-intrusive capability allows `pyuncertainnumber` to interface with multiple software environments, offering extensive compatibility with engineering applications and diverse computational workflows.
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<!-- We interface with many softwares.
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Significance: this provides compatability as interfacing with many engineering applications.
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